Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota : N \to S \otimes_RN$ be the $R$-module homomorphism defined by $\iota(n) = 1 \otimes n$. Suppose that $L$ is any left $S$-module and that $\varphi : N \to L$ is an $R$-module homomorphism from $N$ to $L$. Then there is a unique $S$-module homomorphism $\Phi : S \otimes_RN \to L$ such that $\varphi = \Phi \circ \iota$.
My approach so far is:
Define $\Phi : S \otimes_RN \to L$ by $\Phi(s \otimes n) = s\varphi(n)$.
Show $\Phi$ is well defined.
Show $\Phi$ is in fact an $S$-module homomorphism.
Show $\Phi$ is unique.
I am having trouble with showing that $\Phi$ is well-defined. I suppose $s \otimes n = s' \otimes n'$, it follows by property of cosets that $(s, n) - (s', n') \in S \otimes_RN$. We want to show that $\Phi(s \otimes n) = \Phi((s, n) + S\otimes_RN) = s\varphi(n) = s'\varphi(n') = \Phi(s' \otimes n')$. So if I could show that $s \varphi(n) = s'\varphi(n')$ I would be done with showing well-definedness.
Can someone give me a hint on how to do this? Thanks!